On the Probability That Eisenstein's Criterion Applies to an Arbitrary Irreducible Polynomial
David E. Dobbs, Laura E. Johnson · 2023
A well-known algorithm of Kronecker (cf. [ 1 , pages 126-127]) can be used to determine whether a given polynomial with integral coefficients is irreducible in Q [X]. As it is often time-consuming to implement this algorithm, it has seemed desirable to find sufficient conditions for irreducibility. As described in [ 3 ], the development of such irreducibility criteria has a history exceeding 150 years and remains an active area of research today. Surely, the best-known and oft-cited irreducibility criterion is that named after Eisenstein (cf. [ 1 , page 124]). In fact, in virtually any current textbook on modern algebra, the example given of an irreducible quintic polynomial X 5 + aX + b ∊ Z [X] with Galois group S 5 has its irreducibility established by an appeal to Eisenstein&s;s Criterion. Because such examples may seem somewhat contrived, the question naturally arises as to what the probability is that Eisenstein&s;s Criterion applies to a random polynomial X m + aX + b ∊ Z [X], with m ≥ 2. The purpose of this note is to give sharper meaning to this question and to show that, in a sense, this probability can be bounded independently of m , between 0.2 and 0.3.